### Vector and Matrix

Exercise Express the vectors b = (1, 1, 1), c = (-1, 0, 1), e1 = (1, 0, 0) as linear combinations of the vectors a1 = (1, 2, 3), a2 = (4, 5, 6), a3 = (7, 8, 9). Is the expression (i.e., the coefficients) unique?

Answer We would like to write b, c , e1 in the form x1(1, 2, 3) + x2(4, 5, 6) + x3(7, 8, 9). The computation is similar to the previous exercise. From this exercise, we have

 x1[ 1 ] + x2[ 4 ] + x3[ 7 ] = [ x1 + 4x2 + 7x3 ] 2 5 8 2x1 + 5x2 + 8x3 3 6 9 3x1 + 6x2 + 9x3

Therefore the problem is to solve the system.

 x1 + 4x2 + 7x3 = b1 2x1 + 5x2 + 8x3 = b2 3x1 + 6x2 + 9x3 = b3

For b1 = b2 = b3 = 1 on the right side, we find x1 = -1/3, x2 = 1/3, x3 = 0 is one solution. Therefore b = -1/3 a1 + 1/3 a2. Since the solution is not unique, the expression is not unique. Another possibility is b = -1/6 a1 + 1/6 a3.

For b1 = -1, b2 = 0, b3 = 1 on the right side, we find x1 = 4/3, x2 = 0, x3 = -1/3 is one solution. Therefore c = 4/3 a1 - 1/3 a3. Again the expression is not unique.

For b1 = 1, b2 = 0, b3 = 0 on the right side, there is no solution. Therefore e1 is not a linear combination of a1, a2, a3.